Teleparallel equivalent of Gauss-Bonnet gravity and its modifications

Abstract

Inspired by the teleparallel formulation of general relativity, whose Lagrangian is the torsion invariant $T$, we have constructed the teleparallel equivalent of Gauss-Bonnet gravity in arbitrary dimensions. Without imposing the Weitzenb\"ock connection, we have extracted the torsion invariant ${T}_{G}$, equivalent (up to boundary terms) to the Gauss-Bonnet term $G$. ${T}_{G}$ is constructed by the vielbein and the connection, it contains quartic powers of the torsion tensor, it is diffeomorphism and Lorentz invariant, and in four dimensions it reduces to a topological invariant as expected. Imposing the Weitzenb\"ock connection, ${T}_{G}$ depends only on the vielbein, and this allows us to consider a novel class of modified gravity theories based on $F(T,{T}_{G})$, which is not spanned by the class of $F(T)$ theories, nor by the $F(R,G)$ class of curvature modified gravity. Finally, varying the action we extract the equations of motion for $F(T,{T}_{G})$ gravity.